Crystal approach to affine Schubert calculus

We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$A$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $k$-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb C^n$ enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function $s_\lambda$ for all $|\lambda^\vee|< n$. Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes.Comment: 42 pages; version to appear in IMR

A Comparison of Participant Gains in Attitude and Behavior After Experiencing a Food Safety Curriculum in Traditional and Computer Delivered Environments

Child care providers in Mississippi are required by the Mississippi Health Department to obtain food manager’s training and certification. The TummySafe© program satisfies this requirement and is offered in a self-paced computer delivered version and a traditional classroom version. This research explores participant changes in attitude and self-reported behaviors in the two methods of curriculum delivery as well as the correlation of knowledge change with attitude and self-reported behavior change. A quasi-experimental, pre-test/post-test design was used. Attitude change was not significantly different in the two methods. Traditional participants reported a higher change in self-reported behaviors than computer delivered participants. Both attitude and self-reported behavior change were positively correlated with knowledge gain

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

A crystal on decreasing factorizations in the 00-Hecke monoid

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.Comment: 37 pages; in revision 1 Sections 3.1, 4.3 and 4.4 were updated; in revision 2 the phrase 321-avoiding is replaced by fully-commutative, typos are fixed, reference adde

Enhancing AI High School Student Success: A Work in Progress

This paper describes the first-year activities of a five-year project funded by the U.S. Department of Education as part of the Indian Education Demonstration Grants for Indian Children program. The project brings students, families, the tribal government, and the tribal community together to improve the lives and education of students, as well as their families and community, through a comprehensive change in school culture. The project utilizes a unique, multifaceted approach to offer academic and student support; a four-year Biomedical Science program; Science, Technology, Engineering, and Math (STEM) enrichment; professional development; and community engagement. The overall goal is to assist American Indian (AI) students in making successful transitions to post-secondary educational and career pathways, particularly in STEM fields. The paper describes the work-in-progress and lessons learned, shedding light on current issues in education and encouraging open dialogue about improving the lives of students, families, and communities

Affine charge and the kk-bounded Pieri rule

We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affineanalogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation ofthe Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting

Flag Gromov-Witten invariants via crystals

We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators

Fusion coefficients

Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Eğecioğlu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and Shimozono. We discuss how this approach may work in general